This riddle is subjected to precommisioning test of an instrumentation system in hypothetical industrial plant. Replacing of electrical parameter with other physical parameters is possible, so in addition electrical engineers everyone who involved linear systems can think about this problem and share their knowledge and experience.
The instrument device which shall be checked, includes one current input (0-60 mA ) and one indication pointer that moves proportional to input currents. Purpose of a.m. mentioned test is investigation of linear characteristics of that equipment functions. Random current inputs at many different temperatures are applied to device and following report is recorded by it.

item

input current (mA)

pointer replacing (cm)

temperature (c.d.)

1

7

5

3

2

14

10

6

3

21

15

9

4

28

20

12

5

42

30

18

6

49

35

21

Contractor representative affirms according to above test result, function of device is linear. But representative of client says " those results only prove one of two properties of linear system and superposition property shall be proved too".
Contractor representative believes it is not necessary and " f(ax)=af(x) " is enough to prove systems linear properties, because we can write:

f (ax) = a f (x) eq. no.1

and a=b+c

f (b+c)x = (b+c) f (x) eq. no.2

also f (bx)+(cx) = b f (x) + c f (x)

According to eq.no.1 we can write:

b f (x) = f (bx) , c f(x) = f (cx)

Therefore we can write:

f [(bx)+(cx)> = f (bx) + f (cx)

Because both of b & c quantities are individual terms, therefore (X1= bx) & ( X2=cx) are individual and we can write:

f ( X1+X2 ) = f (X1) + f (X2) eq. no.3

He continues :
The above equation intends superposition property of system.So we prove linear property of function device only with the aid of equation no.1.